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INTERPOLATION METHODS TO ESTIMATE EIGENVALUE
DISTRIBUTION OF SOME INTEGRAL OPERATORS
E. M. EL-SHOBAKY, N. ABDEL-MOTTALEB, A. FATHI, and M. FARAGALLAH
Received 20 August 2002
We study the asymptotic distribution of eigenvalues of integral operatorsTkdefined by kernelskwhich belong to Triebel-Lizorkin function spaceFσ
pu(Fqvτ )by using the factoriza-tion theorem and the Weyl numbersxn. We use the relation between Triebel-Lizorkin space Fpuσ (Ω)and Besov spaceBpqτ (Ω)and the interpolation methods to get an estimation for the distribution of eigenvalues in Lizorkin spacesFσ
pu(Fqvτ ).
2000 Mathematics Subject Classification: 46B20, 47B10.
1. Lizorkin kernels. We will use the following notation:lp,q,Spq(x),Bpqs , andFpqs to de-note Lorentz sequence space, Schatten class, Besov function space, and Triebel-Lizorkin function space, respectively. Byπp,sn, andxnwe denotep-summing norms,s-number function, and Weyl numbers, respectively, see [2,4,5].
Theorem1.1(see [1]). Lets∈R,p∈(0,∞)and letq,u,v∈(0,∞]. ThenBs
pu⊂Fpqs ⊂ Bs
pv if and only if
0< u≤min(p,q), max(p,q)≤v≤ ∞. (1.1)
That is, if and only if0< u≤q≤v.
Proposition1.2(see [4]). LetΦ∈[Bσ
pu(0,1),X]andr=max(p,u). Then
Φop:a →
Φ(·),a (1.2)
(whereΦop is an approximable operator fromXintoBpuσ (0,1)) defines an absolutely
r-summing operator fromXintoBσ
pu(0,1). Moreover,
Φop|πr≤Φ|
Bpuσ ,X. (1.3)
We restate the previous proposition in the following form in the case of Triebel-Lizorkin spaceFσ
pu(Ω).
Proposition1.3. LetXbe a Banach space,Ω⊂RN a bounded domain,σ >0, and 1≤p <∞. LetΦ∈Fσ
pu(Ω;X)andr=max(p,u). Then
Φop:x →
defines an absolutelyr-summing operator fromXintoFσ
pu(X). Moreover,
πrΦop≤ Φp,u,σ;Ω,X. (1.5)
Proof. Givenx1,...,xn∈X, Jessen’s inequality [4] yields
Ω
n
i=1
Φ(ξ),xir
p/r dξ 1/p ≤ n
i=1 Ω
Φ(ξ),xip dξ
r /p
1/r . (1.6) Therefore, n
i=1
Φ(·),xir
1/r Lp ≤ n
i=1
Φ(·),xir Lp
1/r
≤ ΦLpxiπr. (1.7)
Applying this result to∆m
τΦ, we obtain
n
i=1 ∆m
τΦ(·),xir
1/r Lp
≤∆m
τΦLpxiπr. (1.8)
Hence,
Ω
n
i=1
τ−σ∆m
τΦ(·),xiLp
r u/r dτ τ 1/u ≤ n
i=1 Ω
τ−σ∆m
τΦ(·),xiLp
ur /u
1/r
≤ Ωτ
−σ∆m τΦ udτ τ 1/u Lp xi
πr.
(1.9)
Finally, we conclude from the preceding inequalities that
n
i=1
Φ(·),xir
1/r
Fpuσ
≤ n
i=1
Φ(·),xir
1/r Lp + Ω n
i=1
τ−σ∆m
τΦ(·),xiLp
r u/r dτ τ 1/u ≤ ΦLp+
Ωτ
−σ∆m τΦ udτ τ 1/u Lp
= ΦFpuσ xiπr.
(1.10)
Corollary1.4(see [2]). LetXandYbe Banach spaces,2≤p <∞, andT∈πp,2(X,Y ).
ThenT∈Sx
p,∞(X,Y ), and for anyn∈N,
xn(T )≤n−1/pπp,2(T ). (1.11)
We are interested in the following theorem.
Theorem1.5(see [3]). Let1≤p≤max(2,q)≤ ∞. Then
xn
Ip,qm :lmp →lmq
n1/q−1/p for1≤p≤q≤2, n1/2−1/p for1≤p≤2≤q <∞, 1 for2≤p≤q <∞.
(1.12)
Theorem1.6((multiplication theorem) [4]). If1/p+1/q=1/rand1/u+1/v=1/w, then
Spu(x)◦Sqv(x)⊆Sr w(x). (1.13)
Theorem1.7((eigenvalue theorem) [2]). Let0< p <∞,0< q≤ ∞, and letXbe a Banach space. Then any operatorT ∈L(X)which has Weyl numbers(xn(T ))∈lp,q, T ∈Sp,q(x)(X)is a Riesz operator, the eigenvalue sequence of which is inlp,q, and the
following inequality holds
λn(T )
p,q≤cxn(T )p,q. (1.14)
2. Eigenvalue theorem for Lizorkin kernels. The following theorem contains the main result of this note.
Theorem 2.1. LetΩ⊂RN be a bounded domain, 1≤p,q,u,v <∞, andσ+τ > N(1/p+1/q−1). Definerby1/r=(σ+τ)/N+1/q+, whereq+=max(q,2). Then the eigenvalues of any kernelk∈Fσ
pu(Ω;Fqvτ (Ω))belong to the Lorentz sequence spacelr ,p
with
λn(k)
n∈Nlr ,p≤ckFpuσ (Fqvτ ). (2.1)
The constantcdepends only on the indices andΩ.
Proof. First, we assume thatp≤q.
We will show that there exists an imbedding map id :Fσ
pu(Ω)Fqvτ (Ω)and then es-timate its Weyl numbersxn(id). We factories an imbedding map id :Fpuσ (Ω)Fqvτ (Ω) with the help of mapsAandBsuch that
id=B◦idl◦A,
Fσ pu(Ω)
id
A
Fτ qv(Ω)
lm p(Ω)
idl
lmq(Ω). B
This means that
xn(id)≤ AxnidlB (2.3)
if we are able to estimateAandBsuitably; from [6], operatorsAandBare defined exactly as they are in [4], and assume thatΩcontains the unit cube inRN and divide the unit cube in the usual way into 2jNcongruent cubes with side length 2−j.
From [1], we have
A ≤c12−j(σ−N/p), B ≤c22j(−τ−N/q )
. (2.4)
Substituting (2.4) in (2.3), we get
xn(id)≤c2−j(σ+τ)+jN(1/p−1/q
)
xn
idl. (2.5)
ByTheorem 1.5, we have
xnid :Fpuσ (Ω)Fqvτ (Ω)
< n−ρ, (2.6)
where
ρ=σ+τ
N +
0, if 1≤p≤q≤2, 1
2− 1
q, if 1≤p≤2≤q <∞,
1−1
p− 1
q, if 2≤p≤q<∞,
(2.7)
andn=2Nj. Hence,
id∈S1(x)/ρ,∞Fpuσ (Ω)Fqvτ (Ω)
. (2.8)
To estimate the Weyl number ofTkinFqvτ (Ω), we use the factorization
Fτ qv(Ω)
T→k Fσ
pu(Ω) id
Fτ
qv(Ω). (2.9)
ByProposition 1.3, k∈Fσ
pu(Ω;X)implies that Tk:XFpuσ (Ω)is p-summing. By Corollary 1.4, we have
xn
Tk:Fqvτ (Ω) →Fpuσ (Ω)
≤πp
Tk
n−1/max(p,2)≤c1kFpuσ (X)n−1/max(p,2), (2.10)
that is,
Tk∈Ss,∞(x)Fpuσ (Ω),Fqvτ (Ω)
, s=max(p,2). (2.11)
In the case whenp > q, then we have
k∈Fσ pu
Ω;Fτ qv(Ω)
⇒k∈Fqσu
Ω;Fτ qv(Ω)
. (2.12)
In this way the second case is reduced to the first one.
So, we have shown that the mapk→id◦Tk, which assigns to every kernel the corre-sponding operator, acts as follows:
op :Fσ pu
Fτ qv
→S(x)
r ,∞Fqvτ (Ω)
. (2.13)
This result can be improved by interpolation. To this end, choosep0,p1, andθsuch
that 1/p=1−θ/p0+θ/p1. We now apply the formula
Fσ p0u(E),F
σ p1u(E)
θ,p=Fpuσ (E), E=Fqvτ . (2.14)
Then, using interpolation as in [2], where 1/r=1−θ/r0+θ/r1, hence
S(x) r0∞,S
(x) r1∞
θ,p⊆Sr p(x). (2.15)
Hence the interpolation property yields
op :Fpuσ
Fqvτ
→Sr ,p(x)
Fqvτ (Ω)
. (2.16)
By the eigenvalue theorem (Theorem 1.7), we therefore obtain(λn(k))∈lr ,p.
Theorem2.2(eigenvalue theorem for Sobolev kernels). Let1≤p <∞,1< q <∞,
1/r=m+n+1/q+, andw=min(q,2). Then
k∈Wm
p(0,1),Wqn(0,1)
⇒λn(k)
∈lr ,w. (2.17)
Proof. See [4].
The following example proves that our result improves the previous theorem of [4].
Theorem2.3. LetΩ⊂RN be a bounded domain,1≤p,q,v <∞, andτ >0with τ > N(1/p+1/q−1), p≤v, and1/r:=τ/N+1/max(2,q). Then the eigenvalues of any kernelk∈Lp(Fqvτ )belong to the Lorentz sequence spacelr ,vwith
λn(k)
n∈Nlr ,v≤ckLp
Fqvτ
. (2.18)
Proof. We may assume thatp≤q. Then, reasoning similarly as in the proof of Theorem 2.1, it follows that the mapk→Tk, which assigns to every kernel the corre-sponding operator, acts as follows:
op :Lp
Fqvτ
→Sr ,∞(x)Lp(Ω)
This result can be improved by interpolation. To this end, we apply the imbedding
Lp,
E0,E1
θ,m
⊆Lp,E0
,Lp,E1
θ,m, p < m, (2.20)
to the interpolation couple(Fτ0 q,v0,F
τ1
q,v1). The interpolation property now implies that
op :Lp
Fτ qv
→S(x) r ,v
Lp(Ω)
. (2.21)
By the eigenvalue theorem (Theorem 1.7), we therefore obtain(λn(k))∈lr ,v.
Example2.4. (1) In this example, we will indicate a special case of the Lizorkin space Fσ
pu(RN). When 1< p <∞ands∈N0, then
Fs p,2
RN=Ws p
RN (2.22)
are the classical Sobolev spaces.
We compare this case withTheorem 2.2. We find that
k∈Wpσ
Wqτ
⇒λn(k)∈lr ,w, (2.23)
wherew=min(q,2), and
k∈Fσ pu
Fτ qv
⇒λn(k)
∈lr ,p. (2.24)
We conclude that ifp < w=min(q,2), 2≤q <∞, 1< p <2, then
lr ,w⊂lr ,p, (2.25)
that is,
λn(k)
n∈Nr ,p≤λn(k)
n∈Nr ,w. (2.26)
(2) We compare
k∈Wσ p
Wτ q
⇒λn(k)∈lr ,w, (2.27)
wherew=min(q,2), with
k∈Lp
Fqvτ
⇒λn(k)
∈lr ,v, (2.28)
wherep≤v. We conclude that ifv < w=min(q,2), 2≤q <∞, 1< p <2, then
lr ,w⊂lr ,v, (2.29)
that is,
λn(k)
n∈Nr ,v≤λn(k)
References
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[2] H. König,Eigenvalue Distribution of Compact Operators, Birkhäuser Verlag, Basel, 1968. [3] R. Linde,s-numbers of diagonal operators and Besov embeddings, Rend. Circ. Mat. Palermo
(2) Suppl. (1985), no. 10, 83–110, Proc. 13th Winter school.
[4] A. Pietsch,Eigenvalues ands-Numbers, Mathematik und ihre Anwendungen in Physik und Technik, vol. 43, Akademische Verlagsgesellschaft Geest and Portig K.-G., Leipzig, 1987.
[5] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing, Amsterdam, 1978. [6] ,Theory of Function Spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag,
Basel, 1992.
E. M. El-Shobaky: Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt
E-mail address:[email protected]
N. Abdel-Mottaleb: Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt
A. Fathi: Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt
E-mail address:[email protected]
